EBK SYSTEM DYNAMICS
EBK SYSTEM DYNAMICS
3rd Edition
ISBN: 8220100254963
Author: Palm
Publisher: MCG
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Chapter 9, Problem 9.32P
To determine

The expression for the steady state voltage vo (t).

The steady state response for the given function is given by,

EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  1

Given:

The square wave input for the given circuit is given as,

EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  2

Fourier series approximation and is given by

EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  3

Transfer function = EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  4 Where R = 600 EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  5 and C EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  6

Concept Used:

We first calculate EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  7 magnitude of the given transfer function, phase angle, bandwidth, and magnitude of phase angles of respective frequencies and finally calculate the expression for steady state response.

Calculation:

The Fourier series approximation is given by,

EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  8

The transfer function of series RC circuit is given by,

EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  9

The expression for magnitude of transfer function of the given system is,

EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  10 (1)

Substituting for R = 103EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  11 and C = EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  12 in equation (1) we get,

EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  13

EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  14

The phase angle is given by

EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  15 .... (3)

Substituting for R = 103EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  16 and C = EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  17 in equation (3) we get,

EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  18

EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  19 .... (4)

The bandwidth EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  20 for the given system should lie between 0 and EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  21EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  22

EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  23 (5)

Substituting R = 103EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  24 and C = EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  25 in equation (5) we get,

EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  26

EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  27 1000 rad/s

As EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  28 is greater than 1000 rad/s which is outside the required bandwidth, we consider only 0 and 120 EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  29 for frequency values.

The magnitude value for the frequencies 0and 120 EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  30 is,

From equation (2) we know that

EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  31

Substituting for EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  32 0 and 120 EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  33 respectively we get,

EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  34

EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  35 (6)

EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  36

On simplifying we get

EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  37 0.9357.... (7)

From equation (4) we have EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  38

Now calculating the phase angles for corresponding frequencies 0, 240 EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  39 and 480 EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  40 respectively we get,

EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  41

EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  42

EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  43

EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  44

EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  45

EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  46

The steady state voltage for the given Fourier series,

EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  47 is given by,

EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  48

EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  49

EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  50

Conclusion:

Therefore, the steady stat response for the given function is given by

EBK SYSTEM DYNAMICS, Chapter 9, Problem 9.32P , additional homework tip  51

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